Tangent Law

Category : JEE Main & Advanced

When a small magnet is suspended in two uniform magnetic fields

B
and
BH
which are at right angles to each other, the magnet comes to rest at an angle
θ
 with respect to
BH
.

In equilibrium

MBHsinθ
=MBsin(90o−θ)
⇒
B=BHtanθ.
This is called tangent law.

Tangent Galvanometer
Category : JEE Main & Advanced

It consists of three circular coils of insulated copper wire wound on a vertical circular frame
made of nonmagnetic material as ebonite or wood. A small magnetic compass needle is pivoted
at the centre of the vertical circular frame. When the coil of the tangent galvanometer is kept in
magnetic meridian and current passes through any of the coil then the needle at the centre gets
deflected and comes to an equilibrium position under the action of two perpendicular field : one
due to horizontal component of earth and the other due to field (B) set up by the coil due to
current.
In equilibrium

B=BHtanθ
where
B=μ0ni2r
; n = number of turns, r = radius of coil, i = the current to be measured,
θ=
angle made by needle from the direction of
BH
in equilibrium.

Hence

μ0Ni2r=BHtanθ
 
⇒
i=ktanθ
where
k=2rBHμ0N
is called reduction factor.  

Deflection Magnetometer
Category : JEE Main & Advanced

It's working is based on the principle of tangent law. It consists of a small compass needle,
pivoted at the centre of a circular box. The box is kept in a wooden frame having two meter scale
fitted on it's two arms. Reading of a scale at any point directly gives the distance of that point
from the centre of compass needle.
(1) Tan A position : In this position the magnetometer is set perpendicular to magnetic
meridian. So that, magnetic field due to magnet, is in axial position and perpendicular to earth's
field. Hence

BHtanθ=μ04π.2Mr(r2−l2)2
or
BHtanθ=μ04π.2Mr3

(2) Tan B position : The arms of magnetometer are set in magnetic meridian, so that the
magnetic field due to magnet is at it's equatorial position. Hence

BHtanθ=μ04π.M(r2+l2)3/2
or
BHtanθ=μ04π.Mr3

(3) Comparison of magnetic moments : According to deflection method

M1M2=tanθ1tanθ2

According to null deflection method

M1M2=(d1d2)3
 

Vibration Magnetometer
Category : JEE Main & Advanced

Vibration magnetometer is used for comparison of magnetic moments and magnetic fields. This
device works on the principle, that whenever a freely suspended magnet in a uniform magnetic
field, is disturbed from it's equilibrium position, it starts vibrating about the mean position.

Time period of oscillation of experimental bar magnet (magnetic moment M) in earth's magnetic
field
 (BH)
is given by the formula.

T=2πIMBH−−−−−−√

; where,
I=
moment of inertia of short bar magnet
=wL212
(w = mass of bar magnet)

(1) Determination of magnetic moment of a magnet :  The experimental (given) magnet is put
into vibration magnetometer and it's time period T is determined. Now

T=2πIMBH−−−−−−√⇒M=4π2IBH.T2

(2) Comparison of horizontal components of earth's magnetic field at two places

T=2πIMBH−−−−−−√

; since I and M of the magnet are constant,

So

T2∝1BH⇒(BH)1(BH)2=T22T21

(3) Comparison of magnetic moment of two magnets of same size and mass

T=2πIM.BH−−−−−−√

; Here I and BH are constants. 

So

M∝1T2
 
⇒
M1M2=T22T21

(4) Comparison of magnetic moments by sum and difference method

Sum position

Net magnetic moment

Ms=M1+M2

Net moment of inertia

ls=l1+l2

Time period of oscillation of this pair in earth's magnetic field

(BH)

Ts=2πIsMsBH−−−−−−√

=2πI1+I2(M1+M2)BH−−−−−−−−−−−−√

              ....(i)

Frequency 

νs=12π(M1+M2)BHIs−−−−−−−−−−−−√

Difference position  

Net magnetic moment

Md=M1+M2

Net moment of inertia

Id=I1+I2
and 

Td=2πIdMdBH−−−−−−√

=2πI1+I2(M1−M2)BH−−−−−−−−−−−−√

   ....(ii)

and

νd=12π(M1+M2)BH(I1+I2)−−−−−−−−−−−−−√

. From equation (i) and (ii) we get

TsTd=M1−M2M1+M2−−−−−−−−√

⇒
M1M2=T2d+T2sT2d−T2s=ν2s+ν2dν2s−ν2d

(5) To find the ratio of magnetic field : Suppose it is required to find the ratio

BBH
where B is the field created by magnet and
BH
is the horizontal component of earth's magnetic field.

To determine

BBH
a primary (main) magnet is made to first oscillate in earth's magnetic field
(BH)
alone and it's time period of oscillation (T) is noted.

T=2πIMBH−−−−−−√

and frequency
 ν=12πMBHI−−−−−−√

Now a secondary magnet placed near the primary magnet so primary magnet oscillate in a new
field with is the resultant of B and BH and now time period, is noted again.

T′=2πIM(B+BH)−−−−−−−−−−√

or

ν′=12πM(B+BH)I−−−−−−−−−−√
⇒
BBH=(ν′ν)2−1
 

Magnetic Materials
Category : JEE Main & Advanced

  On the basis of mutual interactions or behaviour of various materials in an external magnetic

field, the materials are divided in three main categories. (1) Diamagnetic materials :
Diamagnetism is the intrinsic property of every material and it is generated due to mutual
interaction between the applied magnetic field and orbital motion of electrons. (2) Paramagnetic
materials : In these substances the inner orbits of atoms are incomplete. The electron spins are
uncoupled, consequently on applying a magnetic field the magnetic moment generated due to
spin motion align in the direction of magnetic field and induces magnetic moment in its direction
due to which the material gets feebly magnetised. In these materials the electron number is odd.
 (3) Ferromagnetic materials : In some
materials, the permanent atomic magnetic moments have strong tendency to align themselves
even without any external field.  These materials are called ferromagnetic materials. In every
unmagnetised ferromagnetic material, the atoms form domains inside the material. Different
domains, however, have different directions of magnetic moment and hence the materials remain
unmagnetised. On applying an external magnetic field, these domains rotate and align in the

direction of magnetic field.

(4) Curie Law : The magnetic susceptibility of paramagnetic substances is inversely proportional
to its absolute temperature i.e.

χ∝1T
⇒
χ∝CT
;  where C = Curie constant, T = absolute temperature. On increasing temperature, the magnetic
susceptibility of paramagnetic materials decreases and vice versa. The magnetic susceptibility of
ferromagnetic substances does not change according to Curie law.

(5) Curie temperature

(Tc)
: The temperature above which a ferromagnetic material behaves like a paramagnetic material is
defined as Curie temperature
(Tc)
. or The minimum temperature at which a ferromagnetic substance is converted into
paramagnetic substance is defined as Curie temperature. For various ferromagnetic materials its
values are different, e.g. for Ni,
TCNi=358oC
for Fe,
TCFe=770oC
for          CO,
TCCO=1120oC
At this temperature the ferromagnetism of the substances suddenly vanishes.

(6) Curie-weiss law : At temperatures above Curie temperature the magnetic susceptibility of
ferromagnetic materials is inversely proportional to
(T−Tc)

i.e.
χ∝1T−Tc
⇒
χ=C(T−Tc)
Here
Tc=
Curie temperature  
χ−T
curve is shown (for Curie-Weiss Law)

For ferromagnetic materials, by removing external magnetic field i.e.

H=0
. The magnetic moment of some domains remain aligned in the applied direction of previous

magnetising field which results into a residual magnetism. The lack of

retracibility as shown in figure is called hysteresis and the curve is known as hysteresis loop.

(1) Retentivity : When H is reduced, I reduces but is not zero when

H=0
. The remainder value OC of magnetisation when
H=0
is called the residual magnetism or retentivity. The property by virtue of which the magnetism (I)
remains in a material even on the removal of magnetising field is called Retentivity or Residual
magnetism.
(2) Corecivity or corecive force : When magnetic field H is reversed, the magnetisation
decreases and for a particular value of H, denoted by
Hc
, it becomes zero i.e.,
Hc=OD
when
l=0
. This value of H is called the corecivity. Magnetic hard substance (steel)
→
High corecvity Magnetic soft substance (soft iron)
→
Low corecivity

(3) When field H is further increased in reverse direction, the intensity of magnetisation attains
saturation value in reverse direction (i.e. point E) (4) When H is decreased to zero and changed
direction in steps, we get the part EFGB. Thus complete cycle of magnetisation and
demagnetisation is represented by BCDEFGB. This curve is known as hysteresis curve  
Comparison between soft iron and steel
Soft iron Steel

The area of hysteresis loop is less (low energy loss) The area of hysteresis loop is large (high
energy loss)
Less relativity and corecive force More retentivity and corecive force
Magnetic permeability is high Magnetic permeability is less
I and I and
χ χ
both  are high both  are low
It magnetised and demagnetised easily Magnetisation and demagnetisation is
not easy
Used in dynamo, transformer, electromagnet tape Used for making permanent magnet.
recorder and tapes etc. 
  Comparative study of magnetic materials
Property Diamagnetic Paramagnetic Ferromagnetic
substances substances substances
Cause of Orbital motion of Spin motion of Formation of domains
magnetism electrons electrons
Explanation of On the basis of orbital On the basis of spin On the basis of domains
magnetism motion of electrons and orbital motion of formed
electrons
Behaviour In a These are repelled in These are feebly These are strongly
non-uniform an external magnetic attracted in an external attracted in an external
magnetic field field i.e. have a magnetic field i.e., magnetic field i.e. they
tendency to move from have a tendency to easily move from low to
high to low field move from low to high high field region
region. field region

State of These are weekly These get weekly These get strongly
magnetisation magnetised in a magnetised in the magnetised in the
direction opposite to direction of applied direction of applied
that of applied magnetic field magnetic field
magnetic field
When the Liquid level in that Liquid level in that Liquid level in that limb
material in the limb gets depressed limb rises up rises up very much
form of liquid is
filled in the U-
tube and placed
between pole
pieces.
On placing the The gas expands at The gas expands in the The gas rapidly expands
gaseous right angles to the direction of magnetic in the direction of
materials magnetic field. field. magnetic field
between pole
pieces
The value of B<B0 B>B0 B>>B0
magnetic (where
induction B B0
is the magnetic
induction in vacuum)
Magnetic Low and negative Low but positive c » 1 Positive and high
susceptibility |χ|≈1 χ≈102
χ
Dependence of Does not depend on On cooling, these get These get converted into
χ temperature (except Bi converted to paramagnetic materials
on temperature at low temperature) ferromagnetic at Curie temperature
materials at Curie
temperature
 Relative μr<1 μr>1 μr>>1
permeability μr=102
(μr)
Intensity of I is in a direction I is in the direction of I is in the direction of H
magnetisation opposite to that of H H but value is low and value is very high.
(l) and its value is very
low
I-H curves

Magnetic Very low Very low Very high

moment (M) (≈0)
Examples Cu, Ag, Au, Zn, Bi, Al, Mn, Pt, Na, Fe, Co, Ni, Cd,
Sb, NaCl, CuCl2,O2 Fe3O4
H2O and crown glass etc.
air and diamond etc.

Magnetic Maps and Neutral Points

Category : JEE Main & Advanced

(1) Magnetic maps : Magnetic maps (i.e. Declination, dip and horizontal component) over the
earth vary in magnitude from place to place. It is found that many places have the same value of
magnetic elements. The lines are drawn joining all place on the earth having same value of a
magnetic element. These lines form magnetic map.

(i) Isogonic lines: These are the lines on the magnetic map joining the places of equal
declination.

(ii) Agonic line: The line which passes through places having zero declination is called agonic
line.

(iii) Isoclinic lines : These are the lines joining the points of equal dip or inclination.

(iv) Aclinic line : The line joining places of zero dip is called aclinic line (or magnetic equator)

(v) Isodynamic lines : The lines joining the points or places having the same value of horizontal
component of earth's magnetic field are called isodynamic lines.
(2) Neutral points : A neutral point is a point at which the resultant magnetic field is zero. In
general the neutral point is obtained when horizontal component of earth's field is balanced by
the field produced by the magnet.